Fock Space Coupled-Cluster Method for the Ground and Excited States of the NaMg+ Molecular Cation

This study presents a theoretical investigation of the potential energy curves (PECs) and spectroscopic constants of the NaMg+ molecular cation using the intermediate Hamiltonian Fock space coupled-cluster method with singles and doubles within the (2,0) sector [IH-FS-CCSD(2,0)]. The NaMg+ cation has gained scientific interest due to its potential applications in ultracold chemistry, yet it remains experimentally unexplored. We computed 20 lowest-lying PECs and assigned them to 10 dissociation limits. Our results demonstrated high accuracy, and computed curves are smooth and continuous over the entire range of interatomic distances. The study validates the effectiveness of the IH-FS-CCSD(2,0) method in describing PECs of diatomic molecular cations composed of alkali and alkaline earth metals and provides a solid theoretical foundation for further studies of NaMg+ and similar systems.


INTRODUCTION
−6 Since these systems have a positive charge, they can be controlled by an external electric field, which is a significant advantage in experimental studies.Several researchers have explored this kind of molecular cations in ultracold conditions, including studies of Feshbach resonances of LiBa + , 1 simulations of solid-state physics with a KCa + cation, 2 building atom-ion quantum gates using RbBa + , 3 or the collision studies of LiCa + , 4 RbCa + , 5 and RbSr +6 cations, among others.
In this context, the theoretical studies of potential energy curves (PECs) of diatomics are essential as they serve as a groundwork to produce ultracold species in experimental studies.In this work, we studied PECs and spectroscopic constants of NaMg + , which is an example of previously mentioned molecular cations of alkali and alkaline earth metals.The earlier studies of NaMg + are limited and reported only in a few papers in the context of theoretical investigations of its PECs and spectroscopic constants and, as far as we know, there are no experimental studies of the NaMg + molecular cation.In the 1980s, Peyerimhoff and Bruna carried out calculations for the ground and some of the excited states of NaMg + using multireference configuration interaction with singles and doubles (MRCISD). 7,8Then, in 2017, Fedorov et al. 9 studied NaMg + using coupled-cluster (CC) 10−32 and MRCISD methods, and in 2020, S ́miałkowski and Tomza 33 performed the CC calculations for this system, although both works were limited only to the ground state of NaMg + .Finally, in 2020, the group of Berriche 34 conducted an extensive study of NaMg + for a number of excited states using a pseudopotential approach combined with CISD, treating the system as having two electrons moving in the potential of the two atomic cores.NaMg + has not yet been cooled down to ultracold temperatures due to the fact that it does not meet the criteria for the standard procedure of laser cooling. 34Thus, its properties are yet to be determined using experimental methods.It indicates the necessity for comprehensive theoretical studies of this molecular cation.
The molecular cations composed of alkali and alkaline earth metals are systems characterized by having two valence electrons.In this study of PECs, we decided to employ the IH-FS-CCSD (the intermediate Hamiltonian Fock space multireference CC method with singles and doubles) method, applied to the (2,0) sector. 35This strategy has been validated not only in the studies of diatomic molecules composed of alkali metals, e.g., Li 2 , 36 NaLi, 37 Na 2 , 38 and LiRb, 39 but also in the study of the system similar to NaMg + �the LiMg + molecular cation. 40The main advantage of the method is the fact that it is a purely first-principles method, free of any additional parameters optimized for the chosen system.Moreover, IH-FS-CCSD(2,0) is a size-extensive scheme, meaning that the energies of electronic states of the studied system should converge at an infinite distance to the sum of atomic values, a crucial property in studies of PECs for distances far from the equilibrium.We also correlate all electrons of the system.The IH-FS-CCSD(2,0) strategy used in the calculations for neutral molecules (alkali metal dimers) is based on the idea that we choose a doubly ionized system as a reference, producing closed-shell fragments upon dissociation.Then, we perform calculations using the double-electron attachment (DEA) formalism and produce the energies of the ground and excited states of the studied system.Herein, we study the NaMg + molecular cation; therefore, we chose as the reference system the triple-positive ion NaMg 3+

NaMg
Na Mg Afterward, we used the DEA formalism and obtained the energies of the desired NaMg + cation.The strategy presented above allows us to eliminate the problem that is still a challenge for modern computational chemistry: the proper treatment of open-shell fragments, which are produced during the process of homolytic dissociation of a single bond.The use of the restricted Hartree−Fock scheme (RHF) for bond lengths significantly distant from the equilibrium distance is improper.In this case, the unrestricted Hartree−Fock (UHF) or restricted open-shell Hartree−Fock (ROHF) method should be used.The major drawback is their problems with convergence of the HF and post-HF equations.Several solutions were proposed in the literature to address this issue.−56 However, even though the method is considered as "the method of choice" for studying excited states, it lacks the property of size-extensivity.Thus, its use is limited in the studies of PECs.Another more frequently used technique is the MRCI method, which is size-extensive only in its full CI (FCI) variant, not possible to realize for a multielectron system.An alternative is to replace the core electrons with a pseudopotential or an effective core potential (ECP). 34,57,58In this case, the electron correlation is typically considered solely for valence electrons, often limited to the CISD method, which is equivalent to the FCI for two-electron systems.This approach also requires considering some additional parameters representing the potential of core electrons, and these parameters cannot be used universally for any chosen system.The IH-FS-CCSD(2,0) eliminates the above-mentioned problem with open-shell fragments by employing the DEA strategy, which makes it possible to use the RHF function as a reference function for any internuclear distance.Moreover, it is a strictly size-extensive ab initio method with a correlation of all electrons.
In the next section, the IH-FS-CCSD(2,0) method is described in detail.

METHODS
The coupled cluster method 10−32 is based upon the exponential expansion of a reference function where |Ψ⟩�exact wave function, T�cluster operator, and |Φ 0 ⟩�reference function.The presence of the T operator guarantees that the correlation energy is included in the calculations, and this feature is essential to obtain the best possible accuracy with respect to experimental studies.The T operator is written as T T T T ...
T n describes the excitation from the occupied orbitals i, j, ... into unoccupied (virtual) orbitals a, b, ... In our case, T = T 1 + T 2 for the CCSD model.
The principal idea of the Fock space multireference approach 59−74 is to replace the Schrodinger equation where Ψ is the exact wave function defined within the full configurational space, H is the Hamiltonian, and E is the energy, with the equation to obtain only a few eigenvalues out of the whole spectrum to avoid the diagonalization of the H operator in the large configurational space.The H eff in eq 8 is called an effective Hamiltonian.It is defined within a model space M, spanned by the m model determinants Φ I , with the projection operator P P ( ) The orthogonal space is denoted as M ⊥ with projection operator Q (Q = 1 − P).The effective Hamiltonian operator is defined as where the wave operator Ω is used to construct the exact wave function from the model function and the latter is obtained by the action of P on the exact wave function In the Fock space coupled cluster formalism, the Ω operator, known also as a valence universal one, is defined as where S is a cluster operator responsible for excitations from the model space to the orthogonal one and the braces indicate that normal ordering should be applied to each term of the expansion.
The characteristic feature of the Fock space formalism is that the model space is composed of the configurations obtained not only by electronic excitations within the active space but also via electron attachment and ionization processes.The model space is partitioned into sectors denoted, in our case, by (m, 0), which indicate the number of electrons that have been added to the reference system.Hence, the one-dimensional (0,0) sector is The Journal of Physical Chemistry A associated with the ground state, the (1,0) sector with singleelectron-attached states, the (2,0) sector with double electronattached states, etc.The orbital levels are divided into active levels, which change occupation, and inactive levels, for which the occupation is constant for all reference determinants.The other characteristic feature of the FS approach is the hierarchical structure of the coupled cluster solutions.This implies that when we formulate the Fock space problem for this sector (m, 0), then the cluster operator includes all lower-rank sectors.Herein, only the (2,0) sector of the model space will be applied in order to calculate the PECs of the studied molecular cation.A general expression for the H eff operator for this sector at the CCSD level can be written out as with the model space projection operator P (2,0) defined as where Φ αβ denotes the additional electrons placed on the virtual levels α and β and the respective orthogonal space projector Q (2,0) = 1 − P (2,0) .A serious obstacle in implementing the FS-CC method was the so-called intruder state problem. 75,76It causes severe difficulties with convergence.−85 Briefly, the main idea of IH is to introduce a buffer space between the desired states and the others.This is achieved by constructing the H I operator using the H̅ and the cluster operators S known from the (1,0) sector with and the wave operators Y (2,0) and X (2,0) are defined as with It should also be mentioned that the Fock space approach at the (1,0) sector is equivalent to the EOM-CC scheme applied to the electron affinity.This equivalence means that the eigenvalues are identical, while the eigenvectors can be obtained from each other by a simple transformation.Thus, for the (1,0) sector, we have

RESULTS AND DISCUSSION
The IH-FS-CCSD(2,0) 35 calculations were performed using ACES II 86 ver.2.7.0 software package augmented with our own local module.All spectroscopic constants were determined using the 8.0 version of Robert J. Le Roy's LEVEL program. 87In all calculations, the uncontracted ANO-RCC 88 basis set with additional diffuse functions was used�we refer to this basis set as unANO-RCC+.The six additional exponents for the Na atom are as follows: 0.0044652, 0.0017860 for the s shell; 0.0029282, 0.0011713 for the p shell; and 0.0221083, 0.0088459 for the d shell.For the Mg atom, added exponents for the diffuse functions are the following: 0.0062002, 0.0024801 for the s shell; 0.0053322, 0.0021329 for the p shell; and 0.0209515, 0.0083806 for the d shell.The new exponents were determined with the even-tempered scheme, 89 and the ratio between consecutive exponents is equal to 0.4 for both atoms.The augmented basis set yields the correct ordering of the atomic electronic states.The resulting unANO-RCC+ basis set is composed of 275 spherical harmonic polarization functions.The NaMg 3+ cation was chosen as the reference system for IH-FS-CCSD(2,0) calculations.The reference function employed throughout the study was always obtained by the RHF method.All of the electrons were correlated.The active space size for IH-FS-CCSD(2,0)/unANO-RCC+ calculations was set to 83 (i.e., the 83 lowest virtual orbitals chosen as active), resulting in a model space size of 6889.
The IH-FS-CCSD(2,0) method possesses the property that is crucial in the calculations of PECs, i.e., the size-extensivity.This property states that the energies of the electronic states of a system must converge to the sum of their atomic values at an infinite distance.In Table 1, we presented the following computed values: the first column indicates the dissociation limits, the next four columns indicate the atomic/ionic structure and give their energy values, the next one displays the sum of The Journal of Physical Chemistry A these energies, and the last column presents the energy at the corresponding dissociation limit obtained using the IH-FS-CCSD(2,0) method.The energies of the Na + cation were obtained by using the CCSD method (no electrons attached).
For the Na atom and the Mg + cation, IH-FS-CCSD(1,0) (≡EA-EOM-CCSD) was used (the single electron attachment).For the Mg atom, the IH-FS-CCSD(2,0) method was utilized (the double electron attachment).The last two columns show equal values, which proves the size-extensivity of the method.We computed the 20 lowest-lying potential energy curves of the NaMg + molecular cation using the IH-FS-CCSD(2,0)/ unANO-RCC+ method.These PECs were assigned to 10 distinct dissociation limits.In order to improve the visibility of unique curves, we divided them into Figures 1 and 2�the first one depicts the five lowest dissociation limits of NaMg + , and the second one shows the next five ones.The PECs have different colors for each dissociation limit as well as five different point types for each symmetry and multiplicity of the electronic state.In Figure 1, the PECs are limited to 25 Å, and they are limited to 35 Å in Figure 2, but the energies up to 500 Å are available in the Supporting Information.Obtained curves are smooth over the whole range of interatomic distances from the equilibrium to the dissociation limit.
As we mentioned in the Introduction section, only the group of Berriche et al. conducted the study for the significant number of PECs of NaMg + ; thus, we will mostly compare our findings with theirs.In general, the shapes of PECs are similar to those presented in ref 34 along with characteristic undulations and distortions from the typical Morse-like profile.These atypical shapes are the result of avoided crossings of neighboring curves of the same multiplicity and symmetry, which originate from the interactions and charge transfer processes between the electronic states of Na−Mg + and Na + −Mg.The avoidedcrossing positions were compared with those from ref 34 and they show a high coincidence−the average difference is less than 1% (Table 2).
We also find that only one among the 20 lowest-lying PECs has two potential wells, i.e., the 3 3 Π state.It agrees with the curves computed in ref 34.Since the results from ref 34 are the only ones available for all of the electronic states of NaMg + calculated in this work, and also, it is the most recent literature data for this molecular cation, we will discuss our values mostly in comparison with this paper.The agreement between our values and those from ref 34 depends on the spectroscopic constant and on the considered electronic state.
We identified two states as repulsive, i.e., 2 1 Π and 1 1 Δ.As of the first one, it contradicts the findings in refs 8 and 34 but their D e values are very small (81 and 77 cm −1 , respectively); thus, we consider these are possibly plateaus.The second one agrees with ref 34, where this state is also described as the repulsive one.
Generally, we see good agreement in the values of equilibrium distances.The close correspondence of R e implicates a respective agreement in the derived B e values.As for T e values, the agreement is the best and most consistent.The largest error of ∼1.5% is seen for the 1 3 Π state, but for the majority of states, it is less than 1%.
The agreement with ref 34 in the obtained ω e values is diverse.For some states, we see a very close match (e.g., for the X 1 Σ + and 4 3 Σ + states), but for some others, we observe larger differences (cf 2 1 Σ + and 3 3 Σ + ).We have no explanation for these discrepancies, given that the shapes of our PECs are similar to those in ref 34.Analogously larger differences between our results and those of ref 34 occur in the case of the ω e x e constant.The number of states where these values are close is limited, e.g., 4 1 Σ + or 3 1 Π.To sum up, the agreement with ref 34 is in most cases satisfactory, with minor deviations in some electronic states.

CONCLUSIONS
In this study, we explored the PECs and spectroscopic constants of the NaMg + molecular cation using the IH-FS-CCSD(2,0) method and the unANO-RCC+ basis set.To address the challenge of an accurate description of the dissociation of closedshell species into open-shell parts, we employed the doubleelectron attachment formalism and the RHF reference function.For the first time, the fully size-extensive first-principles method with correlation of all electrons was used to calculate PECs for the entire range of interatomic distances.The curves of the 20 lowest-lying electronic states of NaMg + were obtained and assigned to 10 dissociation limits.The PECs were used to determine the spectroscopic constants of NaMg + : equilibrium distances R e , well depths D e , adiabatic excitation energies T e , harmonic frequencies ω e , anharmonicity constants ω e x e , and equilibrium rotational constants B e .Our results demonstrate good agreement with those of earlier theoretical work. 34nce more, we were able to prove the usefulness of the IH-FS-CCSD(2,0) method in the studies of diatomic systems with two valence electrons.−40 The experimental validation of our theoretical results would be a crucial next step, particularly in the context of ultracold chemistry and collision studies.The broad area of potential applications for this type of molecular cation, such as in the investigation of quantum computing, simulations in the solid-state physics, or precise measurements, holds significant importance for future endeavors.
The t ij... ab... amplitudes are the solution of the CC equations obtained by a projection of the H̅ (≡ e −T He T ) operator against excited configurations ⟨Φ ij... ab... |

Table 1 .
Energies of Electronic States at the Dissociation Limit of the NaMg + Molecular Cation Compared to Respective Atomic Energies

The Journal of Physical Chemistry ATable 3 .
Spectroscopic Constants of NaMg +gThe calculated PECs were used to extract the spectroscopic constants of NaMg + , i.e., equilibrium distances R e , well depths D e , adiabatic excitation energies T e , harmonic frequencies ω e , anharmonicity constants ω e x e , and equilibrium rotational constants B e .The results are gathered in Table3and compared with the available theoretical calculations: MRCISD values from refs 7 and 8, CCSDT and MRCI results from ref 9, CCSD(T) value of ref 33, and finally, the results from ref 34 obtained using the pseudopotential-based method.As we mentioned earlier, there are no experimental data on the spectroscopic constants of the NaMg + molecular cation.

Table 3 . continued
The method used in this work, which is IH-FS-CCSD(2,0)/unANO-RCC+, as described in detail in the text.bThemethod used in refs 7 and 8 is MRCISD.cThefirst method used in ref 9 is CCSDT/cc-pCVQZ.dThesecond method used in ref 9 is MRCI/cc-pCVQZ.e The method used in ref 33 is CCSD(T)/aug-cc-pCVQZ. f The method used in ref 34 is based on pseudopotentials.The basis set for sodium was (7s6p5d3f/6s5p4d2f); for magnesium, it was (9s7p5d4f/7s7p4d4f). g D e , T e , ω e , ω e x e , and B e are given in cm −1 ; R e is given in Å.